A lower bound on ground state entanglement between two regions for a free field

In discrete models, such as spin chains, the entanglement between a pair of particles in a chain has been shown to vanish beyond a certain separation. In the continuum, a quantum field φ(x) at a point represents a single degree of freedom, thus at a region of finite size there are infinite separate degrees of freedom. We show that as a consequence, in contrast to discrete models, the ground state of a free, quantized and relativistic field exhibits entanglement between any pair of arbitrarily separated finite regions. We also provide a lower bound on the decay rate of the entanglement as a function of the separation length between the regions and briefly discuss the physical reasons behind this different behaviour of discrete and continuous systems.